Remark 5.4 Importantly, the periodization of the operator \(\mathcal {L}\)  (Eq. REF ) introduces a discontinuity at the boundaries of the integration domain \(x=\pm L_{{\text{tot}}}/2\) . A first way to get rid of this spurious discontinuity is to consider a periodic version of this operator such as \(\mathcal {L}_{{\text{per}}}=iL_{{\text{tot}}}\sin (2\pi x/L_{{\text{tot}}})\) . Doing so, as was explored numerically in Ref. [1]} while solving an equation similar to the evolution given in Eq. REF , the induced solution looses the convenient property of statistical homogeneity, and in particular energy accumulates at the boundaries. To prevent this accumulation of energy at the boundaries, while keeping an approximate statistically homogeneous region of space near the origin, i.e. far from the boundaries, and using the operator \(\mathcal {L}\)  as it is defined in Eq. REF , we propose to use, instead of the force \(f(t,x)\)  defined in Eq. REF , its truncated version
\(f_{{\text{trunc}}}(t,x)=e^{-\frac{ x^2}{ L^2_{{\text{tot}}}/4-x^2}}f(t,x).\) 
